Question: Simplify and expand the following expression: $ \dfrac{p - 1}{2p + 7}+\dfrac{4p + 2}{p + 9} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2p + 7)(p + 9)$ Multiply the first term by $\dfrac{p + 9}{p + 9}$ $ \begin{align*} \dfrac{p - 1}{2p + 7} \times \dfrac{p + 9}{p + 9} & = \dfrac{(p - 1)(p + 9)}{(2p + 7)(p + 9)} \\ & = \dfrac{p^2 + 8p - 9}{(2p + 7)(p + 9)}\end{align*} $ Multiply the second term by $\dfrac{2p + 7}{2p + 7}$ $ \begin{align*} \dfrac{4p + 2}{p + 9} \times \dfrac{2p + 7}{2p + 7} & = \dfrac{(4p + 2)(2p + 7)}{(p + 9)(2p + 7)} \\ & = \dfrac{8p^2 + 32p + 14}{(p + 9)(2p + 7)}\end{align*} $ Now we have: $ = \dfrac{p^2 + 8p - 9}{(2p + 7)(p + 9)} + \dfrac{8p^2 + 32p + 14}{(p + 9)(2p + 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{p^2 + 8p - 9 + 8p^2 + 32p + 14}{(2p + 7)(p + 9)} $ $ = \dfrac{9p^2 + 40p + 5}{(2p + 7)(p + 9)}$ Expand the denominator: $ = \dfrac{9p^2 + 40p + 5}{2p^2 + 25p + 63}$